The Opposite of Numerical Analysis
Des Higham
Department of Mathematics
University of Strathclyde
[email protected]
Numerical Analysis
The Opposite of Numerical Analysis
Des Higham
Department of Mathematics
University of Strathclyde
[email protected]
Overview
General Principles
Greedy Pathlengths on Small World Graphs
Mean Hitting Times in Chemical Reactions
Based on:
A matrix perturbation view of the small world
phenomenon,
D. J. Higham, SIAM Review (SIGEST), 2007.
Greedy pathlengths and small world graphs,
D. J. Higham, Linear Alg. Appl., 2006.
Comparing Hitting Time Behaviour of Markov Jump
Processes and their Diffusion Approximations,
L. Szpruch & D. J. Higham, submitted.
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Discrete versus Continuous
Typical steps in applied/computational maths
1: Discrete particle-based model
2: Continuous model
3: Discrete grid-based approximation
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Discrete versus Continuous
Typical steps in applied/computational maths
1: Discrete particle-based model
2: Continuous model
3: Discrete grid-based approximation
Numerical analysts look at the issue of
discrete versus continuous for 2–3. But similar questions
arise for 1–2.
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Milgram’s Experiment
The Small World Problem
Stanley Milgram, Psychology Today, 1967
“Given any two people in the world,
X and Z,
how many intermediate acquaintance links
are needed before X and Z are connected?”
Led to the phrase 6 degrees of separation
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Watts & Strogatz Model
“Collective dynamics of ‘small-world’ networks”
D. J. Watts and S. H. Strogatz, Nature, 1998
Small World Phenomenon arises if we have
Clustering: neighbours of neighbours tend to be
neighbours, and a
Small World: any two nodes can be connected by a
relatively short path.
Watts and Strogatz model: take a regular lattice and
randomly rewire a few nodes.
Interpolates between local and global connectedness.
The navigation issue was raised in
“Navigation in a small world”, J. Kleinberg, Nature, 2000
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Nearest neighbours: N nodes on a ring
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For each node, independently, add a new
link with probability p
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Ring with uniform shortcuts added
j
i
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Ring with uniform shortcuts added
j
j
i
i
Greedy Pathlength between nodes i and j: number of
steps to get from i to j where, on each step, we go along
the edge that take us closest to j.
Implicit in the work of Kleinberg, (Nature, 2000)
We will use a Markov chain formulation.
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Markov Chain: states
State space: distance from node 0. Size M := ⌊N/2⌋ + 1.
0
1
1
2
2
3
3
i
i
M−3
M−3
M−2
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M−1
M−2
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Markov chain: transition probabilities
0
1
p*
2
p*
p*
3
1
p*
2
p*
p*
0.5(1−sum)
i−1
3
p*
0.5(1−sum)
i
i
M−3
M−3
M−2
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i−1
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M−1
M−2
p* := 2p/N
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Transition and Hitting
Given that the process in is state i at time n, let pij be the
probability that the process will be in state j at time
n + 1.
The transition matrix P ∈ RM×M is lower triangular and P
highly structured.
Let zi denote the expected number of steps to reach
state 0, starting from state i, and
zave
M−1
1 X
:=
zi
M
i=0
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Mean hitting time system



A

z1
z2
..
.
zM−1

 
 
=
 
where A := A1 + A2 + A3 , with . . .
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
1
1
..
.
1





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Mean hitting time system



A

z1
z2
..
.
zM−1


 
 
=
 
where A := A1 + A2 + A3 , with . . .

1
1
..
.
1






1


4p 
A1 := − N 


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Des Higham
1
1
1
1
1
.. . . . .
.
.
.
1 ... ...
1
1
1






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Mean hitting time system



A

z1
z2
..
.
zM−1

 
 
=
 
where A := A1 + A2 + A3 , with . . .


1
1
..
.
1





 −1 1


−1 1
A2 := (1+ 4pN ) 

.. ..

.
.
−1
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
1
1






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Mean hitting time system



A


z1
z2
..
.
zM−1

 
 
=
 
1
1
..
.
1
where A := A1 + A2 + A3 , with . . .



2p 
A3 := N 


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Des Higham






0
5
0
7
0
..
.
..
.
2M−1
0






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Continuum Limit
Guess that zi ≈ z(xi ), where
4p 1
−
N ∆x
Z
x
z(y ) dy +(1+
0
4p
2p 2x
)∆xz′ (x)+ (
+1)z(x) = 1
N
N ∆x
with xi = i∆x, ∆x := 1/(M − 1), and z(0) = 0.
Also
Z 1
zave ≈
z(y ) dy
0
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Key Lemma
Sequence {zi }M−1
i=1 arises when Euler’s method with
stepsize 2/N is applied to
z′′ (x) + p(Nx + 1)z′ (x) = 0,
z(0) = 0, z′ (0) =
N
2
Solution:
z(x) =
q
−
p √
π
N 2N
e
erf
2p
2
q
q
p √
N 2N π
e
erf
2p
2
Np
x
2
q
+
q p
2N
p
2N
,
Ry
2
where erf(y ) := √2π 0 e−t dt is the error function.
Need to prove convergence.
Relative not absolute.
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p = KN , for fixed K , with N → ∞
K=10
1200
N=4000
N=2000
N=1000
1000
z(x)
800
600
400
200
0
0
0.2
0.4
0.6
0.8
1
x
ODE has uniformly bounded global Lipschitz constant, L.
Taylor series plus Gronwall gives
1
sup |zj − z(j∆x)| ≤ C(L) max {|z′′ (x)| + |z′′′ (x)|}
N [0,1]
1≤j≤M−1
Since z′′ (x) and z′′′ (x) are O(N), overall error is O(1).
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p = KN , for fixed K , with N → ∞
Theorem
Mean hitting time zi satisfies
√ !
i 2K
+ O(1)
N
√
π
zi = N √ erf
2 2K
and
zave
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N
=
2K
√
2K π
erf
2
Des Higham
√
2K
2
!
+e
− 12
!
K −1
+ O(1)
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K
N
Reduction ratio: p =
√
zave (K )
2
=
zave (0)
K
√
2K π
erf
2
2K
2
!
+e
1
− K2
!
−1
+ O(N −1 )
Greedy
Global
Mean−field
0.9
0.8
Pathlength
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−2
10
0
2
10
10
4
10
K
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0 < p < 1 constant, with N → ∞
z(x) ≈
q
′
z (x) ≈
p √
N 4N π
e
erf
p
2
N −
e
2
Npx 2 px
−
4
2
√
Np
x
2
+
1
2
q p
N
p=0.5
100
90
80
70
z(x)
60
50
40
30
20
10
N=4000
N=2000
N=1000
0
0
0.2
0.4
0.6
0.8
1
x
x⋆ :=
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s
2 ln N
Np
Des Higham
⇒
max
{|z ′′ (x)|, |z ′′′ (x)|}} = O(N −1 )
⋆
[x ,1]
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0 < p < 1 constant, with N → ∞
Theorem
Mean hitting time zi satisfies
√
1 p 1
π
zi = N 2 p erf N − 2 i 2p + O (log N)2
2 2p
and
zave
√
π
= N p + O (log N)2 .
2 2p
1
2
Note traditional pathlength is like ln N.
(Greedy alg. is exponentially worse than breadth first
search—similar conclusion on a different model by
Kleinberg, 2000.)
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Markov Jump Versus Diffusion
In many applications,including
chemistry,
cell biology,
population dynamics,
epidemiology,
we can model at different levels:
E.g.
CME (Gillespie): what is probability that we have 237
proteins at time t?
CLE: what is probability that the protein concentration
is between 2.7 and 3.1 at time t?
RRE (mass action ODE): what is the protein
concentration at time t?
These modeling regimes ‘converge’ when the population
size increases . . . . . . how do we quantify this?
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c=1
Example: S → ∅, start with 10 molecules
12
10
8
6
4
2
0
0
0.5
1
CME
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1.5
2
CLE
2.5
3
3.5
4
RRE
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c=1
Example: S → ∅, start with 100 molecules
100
90
80
70
60
50
40
30
20
10
0
0
1
CME
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2
3
CLE
4
5
RRE
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Focus here on mean hitting time
b
x
a
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E[T(x)]
Focus here on mean hitting time
a
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x
b
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Markov jump/birth & death process, Z(t)
Discrete states {0, 1, 2, . . . , M}, with 0 and M absorbing:
P (Z(t + h) = i + 1 | Z(t) = i) = Bi h + o(h),
P (Z(t + h) = i − 1 | Z(t) = i) = Di h + o(h),
P (Z(t + h) = i | Z(t) = i) = 1 − (Bi + Di )h + o(h).
Here, B0 = D0 = BM = DM = 0.
Starting at state Z(0) = j, the expected time to be absorbed
into state 0 or M is given by Uj , where
 (B + D )
1
1
−D2











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−B1
(B2 + D2 )
..
.
−B2
..
.
..
.
..
.
..
.
..
.
..
..
Des Higham

.
.
−DM−1
−BM−2
(BM−1 + DM−1 )











U1
U2
U3
.
..
..
.
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

1
  1 

 
  1 

 
  . 
= . 
  . 

 
  .. 
  . 
1

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Numerical Analysis Viewpoint
Linear system can be written, for 1 ≤ i ≤ M − 1
Bi + Di
Ui+1 − Ui−1
(Ui+1 − 2Ui + Ui−1 ) + (Bi − Di )
= −1
2
2
Standard finite differences on the 2 point BVP ODE
B(x) + D(x) ′′
u (x) + (B(x) − D(x)) u ′ (x) = −1,
2
with u(a) = u(b) = 0
Here b − a = M and we have ∆x = 1
Interesting regime is M → ∞
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Diffusion Approximation
SDE:
dy(t) = (B(y(t)) − D(y(t))) dt
p
p
+ B(y(t)) dW1 (t) − D(y(t)) dW2 (t)
Let w (x) := E[T (x)] be the average first time to hit a or b,
given that y(0) = x.
Then w (x) satisfies the same 2 point BVP ODE.
Want to show that this ODE ‘converges’ to the the finite
difference scheme.
Focus on specific examples . . .
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Production from a source
k
∅→X
Bi = k
and
Di = 0
Mean exit times:
Jump process
b−x
k
Diffusion process
1 −e−2x + e−2a −e−2b 2b
2x
e −e
+b−x
k −e−2b + e−2a
2
−e−2x + e−2a
e−2a 2x
2a
a−x +
e −e
+ 1−
−e−2b + e−2a
2
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Convergence: fix a = 0 and let b → ∞
With x = αb for fixed α ∈ (0, 1), we have
|Jump − Diffusion| ≤ Ce−b min{2(1−α),α)}
where C is independent of b.
Example: k = 5, a = 0, α = 21 :
0
10
−2
Absolute Difference in Exit Time
10
−4
10
−6
10
−8
10
−10
10
−12
10
−14
10
0
10
1
10
2
10
b
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Production
cX
∅→X
Bi = i
and
Di = 0
Mean exit times:
Jump process
b−1
1X1
c s=x s
Diffusion process
Z b 2l
1 e−2x − e−2a
e
−2b
−e
dl + ln b − ln x
c e−2b − e−2a
l
x
Z x 2l e−2x − e−2a
e
−2a
ln a − ln x + e
dl
+ 1 − −2b
−2a
e
−e
l
a
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Convergence
With x = αb for fixed α ∈ (0, 1), we have
|Jump (with a = 0) − lim Diffusion| ≤ Cb−2
aց0
where C is independent of b.
Proof Uses the expansions
n
X
1
s
s=1
Z x
= ln n + γ +
1
+ O(n−2 ),
2n
as n → ∞,
et
dt = γ + ln x + o(1), as x → 0,
−∞ t
where the Euler-Mascheroni constant γ = 0.5772 . . ., and
Z x t
e
ex
1
−2
dt =
1 + + O(x ) , as x → ∞.
x
x
−∞ t
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Example, c = 5, a = 10−3 , α = 21
−3
Absolute Difference in Exit Time
10
−4
10
−5
10
−6
10
−7
10
1
2
10
10
3
10
b
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Degradation
cX
X →∅
Bi = 0 and
Di = i
Mean exit times:
Jump process
x
1 X 1
c
s
s=a+1
Diffusion process
Z b −2l
e2x − e2a 1
e
2b
e
dl − ln b + ln x
e2b − e2a c
l
x
Z x −2l e2x − e2a 1
e
2a
+ [1 − 2b
]
ln x − ln a − e
dl
2a
e −e
c
l
a
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Convergence
With x = αb for fixed α ∈ (0, 1), we have
lim lim (Jump − Diffusion) =
aց0 b→∞
− ln 2
c
Proof Uses asymptotic expansions for
Z ∞ −t
e
E1 (x) =
dt,
x > 0.
t
x
Note: the actual hitting times grow like ln(b), so we have
relative convergence like O(1/ ln b).
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Example, c = 5, α = 12 , a = 10−2 , 10−4 , 10−8
0.14
Absolute Difference in Exit Time
0.13
0.12
0.11
0.1
0.09
0.08
0.07
0.06
a=1e−2
a=1e−4
a=1e−8
0.05
0.04
0
5
10
15
20
25
30
b
ln 2
= 0.1386 . . .
5
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Discussion
This approach of expanding exact solutions breaks down
for more complicated scenarios. E.g.
k
cX
∅→X →∅
involves integrals of the incomplete Gamma function.
Is there a general framework for analysing finite difference
schemes in this non-standard context? At best,
convergence is relative not absolute .
Looking at mean hitting times is practically relevant, and
avoids pitfalls that can arise through the SDE breaking down .
E.g. consider the reversible isometry
c X
1 1
X1 ←→
X2
c2 X2
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Take Home Message
Fundamental modelling issues produce some fascinating,
unanswered,
continuous versus discrete
problems that should be attractive to numerical analysts.
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